Cliodynamics: The Jo urnal of Quantitative Hi story and Cultural Evol ution
Corresponding author’s e -mail: [email protected] k
Citation: Roman, Sabin, and Erika Palmer. 2019. “ The Growth and Decline of the Western
Roman Empire: Qu antifying t he Dynamics of Army Size, Territory, and Coinage . ”
Cliodynamics 10 : x – xx .
The Growth and D ecline of the Weste rn Roman
Empire: Qua ntifying the Dynami cs of Army Size,
Territory, and C oinage
Sabin Roman 1 ,2 , Erika Palmer 3
1 Centre for the Study of Existenti al Risk, University of Cambridge
2 Romanian Instit ute of Scien ce and Tech nology
3 Ruralis — Insti tute for Rural and Regional Resea rch
We model the Western Ro man Empire from 500 BCE to 500 CE,
aiming to unde rstand the interdependent dynamics of army size,
conquered territory and the producti on and debas ement of co ins
within the emp ire. The relationships a re represented th rough feed -
back relationships and model led mathematica lly via a dynamical
system, specified as a s et of ordinary differential eq uations. We
analyz e the stability of a subsystem and determine that it is neutrall y
stable. Based on this, we fi nd that to p revent decline, the optimal
policy was to stop debasement and reduce the army size and territor y
during the rule of Marcus A urelius. Given the nature of the stability of
the s ystem and the kind of policie s neces sar y to p revent decline, we
argue tha t a high degree of centralize d control was necessa ry, in line
with basic tenets of structur al -demographic theory.
1. Introduction
Mathema tical and nume rical modelling has increasingly been passing through
disciplinary bounda ries, with quantitative m odels in the social s ciences becoming
more c ommon. Cliody namics is one such recent deve lopment (Turchi n 20 08,
2011), whe rein the integ rati on of quantitat ive methods a nd historical knowledge
brings new insights i nto huma n behavior and soc ia l institutions.
The prese nt study lies wi thin this domain. Our resea rch focus is to quantify
feedback relationships between a set of variables in the many factors at play in the
decline of the Western Rom an Empi re. There are ce nturies’ worth of ex planati ons
for the decline of the Western Roma n Empire. An early starting point for the study
of the Roman Empi re and its decline is the work of G ibb on (177 6). Sin ce then, at
leas t 210 reasons a nd theories have bee n put forth for th e fall of the R oman Empire
(Storey and Storey 2017), a list of which was original ly c ompiled by Demandt
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(1984). In Ap pendix A, we so rt the se possible reasons into different categories and
discuss their via bility as expla nations of col lapse. Inspired by Tainter (1988), we
identify the foll owing categ ories: exogenous causes that are either natural (e.g.,
cala mities), or soc ial (e .g., foreign invaders), endogenous causes that are e ither
alway s pr esent (e.g., wealth differences) or episod ic (e.g., c ivil wars) and other
reasons that have a moral or mystical nature (e .g., egoism, lac k of dignity).
Any one given cause is unlikely to explain why a complex phenomenon, such as
the dec line of a civil izatio n, occurre d. As such, we focus on de termining the
relationships between se veral interr elate d fa ctors, such as the size of the a rmy,
that of the territo ry, a nd the qua lity and qua ntity o f coi ns, a nd how these
dependencies affected them over time. In additi on, the mode l we have built is
mathematic al and computational in nature, which provides a quanti tative
understanding of the d ifferent facto rs in play. Thus, we move fro m the me ntal (or
conceptual ) models listed in Appe ndix A to qua ntitative, precise on es, where the
magnitudes of the variables are known or determined.
The ma ximum time horizon for our m odel is fr om 500 B CE to 500 CE, and we
aim to understan d the inte rdependent dynamics of th e size of the a rmy, the
territorial expanse and the production and debasement of coins within the empire.
These a re the historical data we used, which pertai n to the full lifespan of the
empire or to key periods (suc h as the height of the empir e) and a re representative
of the e ntire western e mpire. The relationships bet ween the se quantities are
represented through feedbac k mec hanisms a nd model led mathematical ly via a
dynamical system specified as a set of ordi nary differe ntial equations. By solving
the s ystem, we ca n che ck to what exten t the histo rical record is recove re d by the
model. While the parameters of the model a re determined so that the model
predictions best fit the data , they are no netheless archaeologicall y meaningful.
Through a stab ility a nalysis, we show tha t a subsyste m of the mo del ha s a
neutrally stable c enter and periodic orbits. Excluding nega tive values for the
variables, trajectories at an y give n di stance fr om the center are possible . The
trajectory most c losely resembling the historical record for the We stern Ro man
Empire is the one of maximum distanc e from the ce nter, re aching ze ro a s the
minimum of the periodic orbit for each variable. Due to th is stability of the system,
we can rec ommend fiscal and milita ry policies that could probably ha ve prolonged
the existence of the empire by seve ral centuries.
Our resul ts connect t o and partially val idate structural - demographic the ory
(SDT) (Goldstone 1991, 2017; Turchin 2003, 2013). Given the nature of the
stability o f the s ystem a nd t he kind of policies necessa ry to preve nt dec line, we
argue tha t a high de gree of c entralized control was necessa ry, in l ine with basic
tenets of SDT. In SDT terminology, the “ state” ha d a s ignificant role to pl ay in the
dynamics of the emp ire, du e to its milita ry pol icies ( which affect army s ize) a nd
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fiscal mea sures (which concer n the debas ement of coins). Thus , of the key aspe cts
of SDT, the most pe rtinent aspe ct with regard t o our mo del is the high degree of
centraliza tion that directed milita ry and monetary policy in the Roman E mpire.
S ocietal collapse can be de fined as “a rap id, significant l oss of an esta blished
leve l of sociopolitical com plexity” (Tainter 19 88). Howev er, there is a wide r de bate
regarding the notion of collapse and how it applies to spe cific ca ses, such as in the
case of the Ma ya (Aimers 2007; Storey and Stor ey 20 17). Furthe rmore, a process,
be it a c ollapse or any other, is rapid only rela tive to certain t imesca les. For the
Weste rn Roma n E mpire, v iews a nd te rminology range from c alling its later
centuries a decl ine and fall (Gibbon 1 776), a collapse (Tainter 1988) or a slow
colla pse (Storey and S torey 2017). We do not en ter into this debate, but given the
data, we b uilt a model tha t integrates se veral factors in a consistent, minimal
fashion and gives results in line with the archaeologica l record.
We do not cl aim to ha ve ac hieved a n all - encompassing theory of the de cline o f
the Western Roma n Empi re. We neit he r co mpare nor d o we address a ny aspe ct
regarding its relationship with the Ea stern Roman Empire. The div ision of the
empire at 395 CE is treated as an exo ge nous fa ctor in the model , a fe ature we
explain in the m odelling section of the paper. Fu rthermore, our modelling effort is
not aimed at inco rporating al l the complexities of Roma n soc iety, the multitude of
sociopolitical aspects, and rel ationships with f oreig n forces and cultures.
In the foll owing se ction, we r eview several existing qua ntitative models that
capture different aspects of the Ro man Empire (e.g., milita ry strategies, travel ).
Next, in se ction 3, we specify the m odel, explaini ng the e quations a nd the fac tors
they ta ke into acc ount. In section 4, we show the fit to the historical record, p rov ide
an approximate analytic solution and, o n the basis of our model, we prese nt
policies tha t would have p revented the decline of the empire. In sec tion 5, we
provide a discussion of the limitations of the model, particula rly with regard to the
role of gol d in the empire. W e then discus s our findings, specifica lly, the feedbac k
mechanisms the mode l sugge sts existe d within Roma n society, how this rel ates to
wh at is histo rically known, a nd how different policies c ould ha ve affe cted the long -
term evoluti on of the empi re. La stly, we concl ude with a summa ry of the paper’s
contribution. In Appe ndix A, we list the 210 possible reasons that Demandt (1984)
found fo r th e dec line of the R oman Emp ire; we class ify them a nd discuss the ir
adequacy as explanations for decli ne or collapse. In A ppendix B, we detail the
mathematic al a spects of the stability a nalysis for the dyn amical syste m we propose
as a model, along with the m ain results of the sen sitivity a nalysis for the
parameters.
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2. Mathematical and Computational Models of the Roman
Empire
While the decline of the Roma n Empire has been stu died for centuries, the mathe -
matical modelling of the p ossible dynamics of and within the empi re ha s only been
attempted in r ecent decades. Several papers have focused on geographic al a spec ts
of the R oman Empire; e ither re lated to mil itary activity ( Stewart 1999; R eVelle and
Rosing 2000; Henning 2 003), travel ( Graham 2006) or c ommerce (Sch eidel 2013,
2014).
Stewart (1999) introduced a graph -theoretic metho d for ap proaching the
problem of how to s ecure the different regi ons of the Roma n Em pire from possible
attack s. ReVelle and Rosing (2000) a nalyzed this mathema tical problem and
possible strategie s i n grea ter depth. Hen ning (2003) e xplored wa ys of reduci ng the
substantial costs of main tain ing legions a nd develope d a new strategy for tackl ing
the problem. While applic able to the Roma n Empi re, these graph -the oretic
considerations are much more general a nd not specific t o the Roman cas e.
Graham (2006) ca rried out a study that focuse s m ore on the Roma n E mpire.
Employing network analys is, he used A ntonine it ineraries in agen t - based sim ula-
tions of information diffusio n along the different routes. A n estimate was obtained
for the time it would take for info rmation to reach different frac tions of t he
population, a nd the findings were partial ly val idated against the density of
inscriptions i n the differe nt regions of the e mpire. Anoth er model that focu se s on
geographical aspects is ORBIS: The Sta nford Geospa tial Network Model of the
Roman World, which simul ates the time and costs associate d to travel via land,
rivers o r s ea in the Roma n Empire in con ditions approximating the state of the
empire a t 200 C E. By using this netw ork model, Scheid el ( 2013) determi nes the
correlation bet ween maximum prices of transporte d goods an d sailing time.
Similarly, with the s ame n etwork m odel, Scheidel (2014) e stimates travel times
and costs within the empire at courier a nd military speeds during the summer and
winter. W hile the se ge ographical mode ls are important f or q ua ntifying aspe cts of
Roman communica tion a nd travel, they a re sta tic in nature , temp orally
constrained to pe riods a fter the f irst cen tury BCE, a t the matur e s tages of the
empire.
The mode l we develop doe s not have any s patial res olution, but ra ther focuse s
on a ggregate d qua ntities and their inte rdependent dy namics over time, aim ing to
understand peri ods of growth and de cline. Work in a si milar spirit has bee n do ne
by Gü ndüz (2002), who aims to fit a power law to the total area versus time during
the growth stages of the Roma n and Ottoma n empi res. A good fit to the historical
record is ac hieved, with e xponents that are relate d to the golden ratio and other
irrational numbers. Whil e numerical resul ts a re presented, no model for d ifferent
dependencies (e.g., feedba ck relationships) between quantities is pro posed.
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
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Sverdrup and col league s (2013) propose a model for analyzing the sc arcity of
resources in the moder n wo rld. Beyond these considera tions, data on the R oman
world is presented a nd a c onc eptual model is pro posed in the for m of a causal loop
diagram. The diag ram speci fies fe edback relationships between different facto rs
(e.g., popula tion, resou rce base , milit ary strength) but no equati ons a re give n to
capture thes e dependencies quantitatively. Yaroshenko et a l. (2015) conducte d a
wavele t analysis of changes in population and ter ritory for the Roman Empire and
the Europe an Union. The a nalysis tak es as input a t im e series a nd deter mines
whether a t eac h point in tim e the syste m was in a chaotic s tate or not, de pending
on the frequency of the wavel et (higher frequency means more chaotic). Howeve r,
beyond the eva luation of th e states a t different po ints in ti me, no explanatio n for
the results based on causal mechanisms is proposed.
Building upon previous rese arch, we ma p feedback mechanisms bet ween the
army size, land exte nt a nd coin pr oduction, which we capture in a s ystem of
ordinary differential e quations. Thus, b y solving this sys tem of e quations, we can
see how thes e aggregate quantities evol ve ove r time a nd c ompare them with the
archae ological record. This all ows us to validate the model in the s ense that it
forms a p ossible set of dyna mics for the real syste m, at least within the range and
scope of the data use d.
3. Model Specification
In this s ection, we fir st outline the model ling me thodology we employe d and then
introduce our model (3.1) – (3.5) for the growth and decl ine of the Western Roman
Empire. Afterwa rds, we disc uss its structu re a nd how th e different ter ms can be
interpreted, a long with the pa rameters and their val ues, which a re given in Ta ble
1.
In the model ( 3.1) – (3.5), we account for the g rowth a nd dec line of the a rmy
size, the l and conque red and the production and debase ment of silve r co ins. The
evolution of the ar my size is estima ted from multiple s ources (Ma cMulle n 1980;
Ward 1990; Roth 1999; Campbell 2006) . The extent of land conquests over time is
from Taagepe ra ( 1979), and the data for the pr oductio n of coins is from Ho pkins
(1980), while debasement informati on is f rom Tainter (1988, 1996). The time span
of the data r egarding the army and l and is 1000 years, from 500 BCE ( -500) to 500
CE (+500) , which dete rmines the maximum time horizo n for th e model. We refer
to the his torical time series, which we aim to re produce as reference mo des (see
Fig ur e 1).
G i ve n t h is da t a , t h e ta s k o f de ve l o p i n g a d y n a mi c al s y s t e ms m od e l c a n b e
understood as constructi ng a set of ordinary differe ntial e quations (ODE s), whose
solutions reproduce the observe d historical trajectories. The system of ODEs gives
th e ra t e o f c ha n g e o f v ar i a bl e s ( e . g . , ar m y s i ze , t e r r it o ry ) , w hi c h a re ty p ic al l y
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Table 1. Pa rameter and initi al values for model (3.1) – (3.5)
Notation
Meaning
Value
𝑝 0
Debaseme nt effect on change s of army size
1142 soldiers/ye ar
𝑝 1
Scaling paramete r for debas ement
0.5
𝑝 2
Army effect on cha nges in area of terr itory
326 km 2 /yea r
𝑝 3
Scaling paramete r for army size
200 000 soldiers
𝑝 4
Debaseme nt effect on cha nges of area of
territory
14 276 km 2 /year
𝑝 5
Territory effec t on changes in debase men t
0.002856
𝑝 6
Scaling paramete r for area of terr itory
2 500 000 km 2
𝑝 7
Net growth rate o f silver rese rves
2.5%
Λ
Army and territo ry decrease after division
55%
𝑡 0
Initial time
- 500
𝑡 𝑑
Time of division
395
𝑥 ( 𝑡 0 )
Initial value of army size
48 850 soldie rs
𝑦 (𝑡 0 )
Initial value of territory area
648 521 km 2
(𝑧 1 𝑧 2
⁄ )(𝑡 0 )
Initial value of silver concentration
17.3%
𝑧 1 ( 𝑡 0 )
Initial amou nt of silver
0.33 million coi ns
𝑧 2 (𝑡 0 )
Initial number of coins
1.9 1 million coi ns
expresse d as linea r and nonlinear c ombinatio ns of the va riables thems elves. Often,
these type of m odels do not ha ve an explicit time de pendence and are c alled
autonomous. Exogenous effec ts, not modelled directly by the sys tem, ca n be
incorporated as t ime-de pendent terms.
Mathe matical models of this type abound in physics, mathe matical biology and
engineering (Str ogatz 2015 ). W ithin the social sciences, a s ubstantial number of
such models ha ve been developed in the field of system d ynamics (Ste rman 2000).
Beyond fitting the observed time series, the model should have a st ructure and
terms tha t are ove rall me aningful fr om a psychol ogical, s ociolo gica l and historical
perspective. When construc ting a n ODE model of a c omplex social system, one
main gu iding p rinciple is th e ide ntification of fe edback mechanisms between the
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
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different varia bles. This allows us to uncove r dependenc ies between variables a nd
to ensure that the relatio nships wi thin the mathem atical model refl ect real
qualitative a nd quantitat ive feature s.
However, developing a mode l tha t fits the historical record, is fa ithful i n
capturing feedback relation ships, has terms a nd paramete r va lu es that ca n be
calibrated a nd interprete d meani ngfully an d is robus t e nough to overcome
uncertainties in the da ta, i s challenging (Sterman 2 000). Theoretical insights,
feedback analys is and modelling principles (such as those employed in population
dynamic s [Turchin 2003]) c an help identify meaningful equations for the model,
but the ta sk of ve rifying th e adequate c ombination of fac tors is laborious . Our
initial aim was to use an aut omated equation discove ry framework, such as SINDy
(Brunton et al . 2016) th at performs linear regression on g radients ( i.e., rates of
change) on combinations of variable s to iden tify ODEs that gene rate trajectories
consistent with the reference modes.
However, the unce rtainty in the da ta a nd its granular nature prevents the
determinatio n of reliable g radient in formation, which the n make s automated
equation d iscovery unfea sibl e. A way to resolve the low resolution problem in the
data is to use a s mooth appr oximation and, in the case of the reference modes for
the army, land and coi n de ba sement, we found tha t the data is well a pproximated
by sine and c osine curves, s uch as in th e solution (4.1). The n, we could i nfer the
system of o rdinary different ial e quations of the subsys tem ( 3.1) – (3.3) , which ha s
an almost linear structure (exce pt for the disc ontinuities ). Thus , we a rrived at part
of the model in a semi-analytic way and did not have to resort to searching i n hi gh -
dimensional fea ture spa ce, which a utomated equation d iscovery frameworks have
to do. We deta il below furthe r aspects of the model.
I n t o t a l , t h e r e a r e f o u r i n d e p e n d e n t e q u a t i o n s i n t h e m o d e l ( 3 . 1 ) – ( 3 . 5 ) ;
specifical ly, only two of the l ast three equations (3.3) – (3.5) are independent. These
latter equations a ccount f or the numbe r of minted silve r coins 𝑧 2 , the amount of
silver use d 𝑧 1 (mea sured in e quivale nt number of pu re s ilver coins) and the ratio
𝑧 1 / 𝑧 2 tha t gives the a verage s ilver content of a coin in ci rculatio n, which is h ow the
debasement of coins is measure d and tracked in the hi storica l data (van Hee sh
2011). Data referring to c oin production and debas ement is ava ilable over a
shorter time span than for the army or l and, an d we cannot gua rantee a ccur a te
results outside the time pe riod for ea ch dataset. W e do not attempt to captur e in
the model the va riety of different coins us ed by the R omans throughout time.
Instead, we a im to a ccount for a n aggregated measure in 𝑧 2 , which reflec ts an
equivale nt tota l value . N evertheless, the repres entative c oins i ncorporated in the
model are mostly c losely refle cted in denarii : see Figure 1 (d).
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
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Let 𝑥 be the army size, 𝑦 the land conque red, 𝑧 1 the amount of s ilver use d in
minting (me asured in c oins) and 𝑧 2 the number of minted coins. We propose the
following dynam ics for these quantit ies:
𝑥 = 𝑝 0 ( 𝑧 1
𝑝 1 𝑧 2 − 1) − 𝜆𝛿 ( 𝑡 − 𝑡 𝑑 ) 𝑥
𝑦 = 𝑝 2 ( 𝑥
𝑝 3 − 1) + 𝑝 4 ( 𝑧 1
𝑝 1 𝑧 2 − 1) − 𝜆𝛿 ( 𝑡 − 𝑡 𝑑 ) 𝑦 ( 3 .2 )
( 𝑧 1
𝑧 2 ) ′ = 𝑝 5 (1 − 𝑦
𝑝 6 ) ( 3.3 )
𝑧 1 = 𝑝 7 𝑧 1 (1 − 𝑦
𝑝 6 ) ( 3 .4 )
𝑧 2 = 𝑧 2 (1 − 𝑦
𝑝 6 ) (𝑝 7 − 𝑝 6 𝑧 2
𝑧 1 ) ( 3.5 )
where the paramete rs 𝑝 0 , … , 𝑝 7 a re as sumed non -negative a nd determined such
that the trajec tories the s ystem generates match as closely as possible to the
reference modes (see Figure 1). The ye ar 𝑡 𝑑 = 395 CE is when the empire divide d
into tw o separate parts for political reasons, with the surface area of the western
em pire de creasing by 𝜆 = 55% (Taa gepera 1979). This c hange is model led as an
exogenous s hock to the syst em, where the la nd exte nt and army s ize are reduc ed
by a fraction 𝜆 .
The fi rst e quation (3.1) o f the model dictates the evolut ion of the army size 𝑥 .
The first contribution to t he rate of change of the a rmy size is give n by the
debasement of the coins in c irculation. We can understa nd this as follows: if a
conquest was s uccessful , the n the influx of silver and other r esource s l ed to
economic b enefits, e. g., red uced taxation t o citizens (Ta inter 1988), which al so
translated in to coins of high purity. This the n encoura ged further con quest, which
remained i n proportion to the returns, as measure d in the quality of the min te d
coins. The operatio ns on 𝑧 1 /𝑧 2 within the parenthesis cha nge the val ue from the
interval [0; 1] to [-1; 1] , so that contractions can be al so be experienced a s e xpected
if the retu rns a re too low and the quality of coins dec lines. The para meter 𝑝 1 se ts
the scal e difference between [-1; 1] and the range of 𝑥 .
The second contri bution to the army is due to the divisio n of the empire at 𝑡 𝑑 =
395. The drivers o f this split were political in nature (Ta agepe ra 1979) a nd could
not be c aptured i n an auto nom ous way (with no expl icit time dependence). Thus,
we m odel the divisio n oc curring at 𝑡 𝑑 = 395 as a n e xogenous change ( or s hock) tha t
reduces the army size and land extent by half. The exoge nous shocks are modelle d
with Dirac delta functions 𝛿 ( 𝑡 − 𝑡 𝑑 ) , ce ntered at the time of the division.
(3.1)
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
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Figure 1 . Compari ng the h istorical reference modes (solid lines) with model
output (da shed lines). The e volution of (a) army size, (b) la nd con quered and (c )
debasement (measured as p ercent age silve r content of coins) as predic te d by the
model. In (d), the predicte d a mount of minted c oins foll ows a similar trend to the
historical record and matches wel l on the downward t rajec tory.
The second e quation (3.2) of the model spec ifies how the land conquered 𝑦
changes. There a re two co ntributing terms, distinct from the exogenous shock
mentioned. First, the army size c ontrib utes to mor e land being co nquered. Second,
the c onquest of more l and t ake s place if there a re suff iciently high retur ns to this
activity, whic h is mea sured by the quality of minted c oins (given by the silver
content), just as for the arm y size.
C ompa ring the parame ters 𝑝 2 and 𝑝 4 , we can see tha t the second ter m, due to
the debaseme nt, has a more significant influence on the ra te of c hange of the land
conquered 𝑦 . By looking a t the solutions (4.1) of the equat ions in subsystem (3.1) –
(3.3), we can see that the first term, due to the army size, amounts to a delay of 𝛿 =
4 years (see Tab le 2) in the expansion of l and compared to the growth of the army.
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
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quality a nd quan tity of the W es tern R oman Empire — can generate trajectories tha t
close ly follow the archaeological r ecord for these quantities. In principle, any
number of models could a chiev e a good fi t to the hist orical data; ho wever, th is does
not gua rantee the model is meaningful from a soc iopolitical or econ omic
perspective or that it is c onservative i n the nu mber of parame ters and
relationships that it hypoth esizes ( in the s pirit of Occ a m ’ s razor). Given the data
we focused on (the re ference modes in Fig ure 1), we have pr ovided what we
intended to be a mi nimal mathematica l structure (in the form of a system of
ordinary differential equatio ns) sufficient t o account fo r the behavior seen in the
time series.
A visible misma tch of the m odel and data is noticeable in the e arly periods in
Figure 1 (c) and ( d). W ith reg ard to Figu re 1 (c), sil ver con centration o f coins sta rt s
at a pproximately 20%, which is c onsistent with the silve r content of an c ient c oins
(Wick ens 1995). Thus , histo rically and acc ording to the model, coins prior to 137
BCE have a lower silver cont ent. He nce, r egarding Fig u re 1 (d), the model output is
an e stimate of the total nu mber of c oins i n circulatio n, not just of h igh silv er
concentration (which is wha t the data show). Coins with highe r silver content are
better preserved, so more reliable data is a vailable for these types of c oins (va n
Heesh 2011). This can explain the mismatch b etwee n the availa ble da ta a nd model
output.
Another possi ble e xplanation is that, as we ca n see in Fig ure 1 (d), a t 100 BCE
several hu ndreds of millions of dena rii we re in ci rculation, and we as sume that the
extent of ec onomic activity was p roportional to availa bility of the coi ns. While
coins of hi gh purity could b e found befor e 200 B CE, this does not n ecess arily mean
their number was r epres entative of the exte nt of e conomic activity at tha t time
(Bowman and Wils on 2009). The model structure , on the othe r hand, remai ns
unchanged, an d i ts output s how s the e quivalent in coin production a nd purity fo r
the overall economic activity even at early points in the empire ’ s history (just as it
also does whe n the a vailability and quality of coins i n th e data is refle ctive of the
economic state of the later e mpire).
If there is a match of model output a nd data, this does not me an the model is
historically a ccurate, but only that is consistent with k nown historical data and
cannot be falsif ied withou t more data. A mo del does not have to be c omplete ly
dismissed if fou nd inconsis tent wi th data , but rather it c an be extended to in -
corporate new ranges and scales, while the previous (va lid) models are re cove red
in certain l imits. F or exam ple, wit hin the time span in Fig ure 1 (a), (b) and (c ), the
subsystem (3.1) – (3.3) ma tches the histor ical record wel l, independe nt of the fit in
Figure 1 (d). The mismatch of model outpu t a nd data may me an tha t the mo del is
not val id in a certai n doma in or tha t the inte rpretation is no longer compati ble with
the data, as may be the c ase with (3.4) – (3.5) and Figu re 1 (d).
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
17
Still, there are ge nuine mo del ling artefac ts i nconsistent with the da ta, within
the scope of this model. I n pa rticular, in Fig ure 1 (c) , after the yea r 400 CE the sil ver
content of c oins increas es, which poin t s to a l imita tion of the model. This i ncon-
sistency arise s from the disconti nuity at 𝑡 𝑑 = 395 in the ar my size a nd land exte nt.
While this could be correcte d by additi onal te rms in the e quation for the ra te of
change of 𝑧 1 / 𝑧 2 , we prefer to k eep the st ructure of the model ( 3.1) – (3.5) sim pler
and allow for g reater transpa rency of the model ’ s limi ts.
How d oes the model structu re relate to known hist orical processes ? As F ig ure
1 shows, the e quations are sufficient to reproduc e the reference m odes well.
However, the struc ture of the mode l goes beyond this go od fit. It implies that th e
value of the cur rency was a s trong deter minant of both ar my g rowth a nd ter ritorial
expansion. Historicall y, it is known that in case of cash shortages (e.g., due to
military ca mpaign c osts), the central authority of the empire would debas e the
coins (Ga rnsey and Sal ler 2 015). Fisca l policy i nvariably affects the re sources of
the army and the amount of l and tha t can be conquered, and similarly, the success
of campaigns affec ts re venue of the em pire a nd the fiscal policies in place
(including coin debase ment) . Thus, the re were impo rtant caus al feedback rela tion-
ships between milita ry costs, territorial e xpansion and monetary issues .
Furthermore, Ta inter (1988) pro poses a theory of socie tal colla p se tha t
surpasses the diff iculties we outlined in Ap pendix A, b uilding upon the idea of
diminishing returns to investments in proble m solving. This is exemplified well by
the Roman Em pire, whose e arly conquests we re very pr ofitable and al lowed for
the elim ination of taxes for the citizens. With the e xpanding territory , military and
administrative costs grew as well. At a c ertain point, further conquests a nd
conflicts pr oved le ss beneficial and e ven amounted to a l oss of res ources, l ike the
wars with the Germa n ic tribes . Maintenance of the em pire ended up having larger
costs tha n revenue, a nd te rritory wa s gradual ly lost. T he Roman currency, the
denarius, was being debase d to expa nd the mo ney sup ply and cover the c osts (at
leas t temporarily) (Tainte r 2000). T hroughout the pe riod 200 – 500 CE, thes e
negative returns manifes ted as the decline of the empire. Hence , at least according
to Tain ter ’s (1988) theory, the fe edback mechanisms that e xisted in the empire
connect di rectly to its o bserved long -term devel opment a nd decl ine. Simil ar
arguments have been put forward rega rding the C hinese dynastic cyc le (Lattimore
1940) and the Ottoman Emp ire (Lewis 1958), as wel l as the Maya (Culbert 1991).
While such relationships are qualitatively known, the model (3.1 ) – (3.5) we
propose p rovides in additi on a prec ise, qua ntitative way to e n capsul ate the se
feedback relationships. Thus, it moves from a mental model to a mathema tical one
(Sterman 2000). F urthermore, e xcept for a discontinui ty at a specific time, the
subsystem (3.1) – (3.3) has a linear structure. Thus, given the va riables that we
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
18
investigate, it is a particula rly simple structure tha t affords sufficient e xplanatory
power to match the historic al record.
Rega rding the mitigation policies, we rema rk tha t the timing for the opti mal
policy is 171 CE, at the midpoint of the rule of Ma rcus Aurel ius. W hat the subsyste m
(3.1) – (3.3) sugge sts is that if at this point the empir e had been split in to two
roughly equal parts and a fiscal policy imple mented to keep silver c ontent at
approximatel y 50%, then the Western Empi re coul d ha ve potentiall y laste d for
much longer in this new sta te. A similar policy was a ctual ly implemented in 395
CE, but only with r egard to t he a rmy s ize and l and exte nt. Raising the silve r c ontent
of the more widel y and highl y circulate d coins was not fe asible anymore, a nd g ol d
had different dynamics of circul ation, eve n if it was more a bundant tha n in earlier
times. In a ddition, the real cha nges we re impleme nted too la te to al ter the
trajectory su bstantially and preve nt dec line. Neverthe less, the me asures take n in
395 CE are in line wi th w hat the phas e portrait of t he subsystem (3.1) – (3.3)
suggests to have been the adequate course of action, namely reduction of stocks to
value s closer to the fixed point. Of cou rse, the model ( 3.1) – (3.5) is an idealization
of reality and it is debatable whethe r the policy it suggests could have worked, but
it does offer an in teresting thought experime nt in this rega rd.
In addition, it is worthwhile to compare the dy namics of the model (3.1) – (3.5) ,
which desc ribes the soc ioeconomic sys tem of the R oman E mpire, w ith the
dynamics for models of socio -e cological system s suc h as Easter Isl and (R oman e t
al. 2017) and the Cl assic May a (Roman et al. 2018). In the case of Easter Is land and
the Cl assic Maya , the collapse is modelle d by a s uper- critical Hopf b ifurcation
where, if the parameter rep resenting the ha rvesting rat e of resourc es per capi ta
excee ds a critical threshold, the system moves from a stable fixed poin t to an
attractive periodic orbit (a li mit c ycle) of l arge amplitude . Once the syste m re ach ed
the l ower limits of the orbit, the coll apse occ urred. The necess ary change to
prevent c ollapse in this case is a change in the harves ting rate of res ources, which
translates in to a lifestyle change for the enti re society and its r elationship t o its
environment.
For the We stern Ro man E mpire, the dynamics for the subsys tem ( 3.1) – (3.3)
shows that neut ral stability a nd changes in parameter va lues are not the course of
action best suited to prevent decline — it is the values of the variables (stocks) that
require tuning to reduc e osc illa tions. Rather than a lifes tyle or cultural change for
all inhabitants, the policy in the case of the s ocioeconomic syste m of the Roma n
Empire is mo re fea sibly i mple mented by centralize d state powers and inte rvention
by e lites. Th us , to ta ckle in sta bility (i n the sense of ter ritorial loss), the state would
need to interve ne with a fisc al and military policy sup ported by elites that a ffects
the wider populatio n but ensures te rritorial integrity an d economic stability (e .g.,
prevention of inflation via d ebasement). Thus, the dynamic s the model (3.1) – (3.5)
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
19
uncovers fo r the Roman Empire rep resent a specific instance of the feedback and
relationships posited by structura l -demographic theory ( Baker 2011). In this
sense, through the exampl e of the socioecono mic system we analy z e, our work
provides partial va lidation of the m o re general framewo rk tha t structu ral -demo-
graphic theory de velops.
6. Conclusion
Hundreds of rea sons have bee n put forward for the fa ll of the Wes tern Roman
Empire ; we have classified them in to the se veral ca tegories in Ap pendix A. Any one
reason is not suff icient to expla in the de cline of such a complex society. W e ha ve
aimed to devel op a n u nderstanding tha t l inks several depe ndent fac tors together.
On the other hand, give n the compl exity of Ro man societ y, it would be unfeasible
to attempt a mo del that covers al l possible features .
Thus, we identified ce rtain key fee dback relationships in the We stern Roman
Empire betwee n aggregated va riables representative for the whole e mpire, for
which we had da ta: the arm y s ize, the area of the territ ory, and the debaseme nt
and quantity of sil ver coins. The main focus was o n un derstanding the evolution
over time of these variables , and we built a syste m of or dinary differe ntial
equations that captures this feedback quantitatively.
The linear structure of a subsystem of the model allowed us to solve par t of the
system of equatio ns analyticall y up to the time of division of the empire at 39 5 CE.
Parameters in the m odel were opt imized to ma tch the his torical record a s close ly
as possible. In general, e ven an optimal choice of parameters does not gua rantee a
good fit. Howeve r, in this case , the structure of a model gives nume rical sol utions
that show a c lose fit to the historical record. F urt he rmore, the parameters a re
archae ologically meaningful, relating to the sca le of the va riables.
A sta bility analys is determine d that the linea r subsystem has a neutral cente r ,
with periodic orbits (f or details , see Appendix B ) . This implies that by mak ing
adequate e xogenous cha nges to the system at the right time s and to the right ex -
tent, the life s pan of the We stern Empire coul d ha ve been increased significan tly.
We found tha t the optimal policy is to rou ghly hal ve the size of the army and
territory and fi x the silver content of coins at 50% at 171 CE, du ring the rul e of
Marcus Aurel ius.
For socio-ecologica l systems, such as Easter Island (Roma n et al. 2017) a nd the
Class ic Maya (Roman et al. 2 018), the coll apse can be modelled as a type of critical
transition, in which a sta ble fixed point cha nges to a n attractive l imit cyc le, which
is an is olate d perio dic o rbit. The critical pa rameter t hat determine s the sus-
tainability or c ollapse of the system is the extrac tion rat e of resources per capita ,
which, i f high enou gh, seve rely de grades the ecosyste m support o f the s ocieties.
For the Weste rn Roma n Empire, the dec line is not due to a critical transition but
Roman and Palmer : Growth and Decline . Cliodynamics 10 : 2 (201 9)
20
an unsus tainable trajecto ry from the beginning, which c ould ha ve bee n c hanged
through a r eduction in the values of key variable s. Such a c hange was attempted in
395 CE, when the empi re di vide d, but the debasement of coins was too seve re to
allow for reve rsion to higher qual ity. In this late r period, gol d coinage became more
common but ha d a differe nt, more restricted circulatio n ( Bransbourgh 2015; Guest
2008) and thus, we argue, d oes not affec t the model results.
We have not solve d the dee p er p roblem of why the W estern R oman Empire
declined, a question that has been posed for centu ries. However, we provide
insight into the i nterlock ing dynamics of so me key aspects of the empire, sub -
stantiated by a quantitative model and anal ysis that offers a precise , math emat-
icall y definite view on the proble m beyond conceptua l models .
Acknowledgme nts
We thank the Gra ntham F oundation for suppo rting this work and the a nonymous
reviewers for their comments and insights .
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